AIR FORCE INSTITUTE OF TECHNOLOGY · 2018. 8. 22. · Current statistical orbit determination (SOD) techniques are able to esti-mate and track trajectories for cooperative spacecraft

SPACE-BASED MANEUVER DETECTIONAND CHARACTERIZATION USING

MULTIPLE MODEL ADAPTIVEESTIMATION

THESIS

Justin D. Katzovitz, 2d Lt, USAF

AFIT-ENY-MS-18-M-268

DEPARTMENT OF THE AIR FORCEAIR UNIVERSITY

AIR FORCE INSTITUTE OF TECHNOLOGY

Wright-Patterson Air Force Base, Ohio

DISTRIBUTION STATEMENT A. APPROVED FOR PUBLIC RELEASE;

DISTRIBUTION IS UNLIMITED

The views expressed in this thesis are those of the author and do not reflect the officialpolicy or position of the United States Air Force, the United States Department ofDefense or the United States Government. This is an academic work and should notbe used to imply or infer actual mission capability or limitations.


SPACE-BASED MANEUVER DETECTION AND CHARACTERIZATION

USING MULTIPLE MODEL ADAPTIVE ESTIMATION

THESIS

Presented to the Faculty

Department of Aeronautics and Astronautics

Graduate School of Engineering and Management

Air Force Institute of Technology

Air University

Air Education and Training Command

in Partial Fulfillment of the Requirements for the

Degree of Master of Science in Astronautical Engineering

Justin D. Katzovitz, BS

2d Lt, USAF

March 2018

DISTRIBUTION STATEMENT A. APPROVED FOR PUBLIC RELEASE;

DISTRIBUTION IS UNLIMITED




Justin D. Katzovitz, BS2d Lt, USAF

Approved:

Joshuah A. Hess, PhD (Chairman) Date

Richard G. Cobb, PhD (Member) Date

Kirk W. Johnson, PhD (Member) Date


Abstract

An increasingly congested space environment requires real-time and dynamic space

situational awareness (SSA) on both domestic and foreign space objects in Earth

orbits. Current statistical orbit determination (SOD) techniques are able to esti-

mate and track trajectories for cooperative spacecraft. However, a non-cooperative

spacecraft performing unknown maneuvers at unknown times can lead to unexpected

changes in the underlying dynamics of classical filtering techniques. Adaptive estima-

tion techniques can be utilized to build a bank of recursive estimators with different

hypotheses on a system’s dynamics. The current study assesses the use of a multiple

model adaptive estimation (MMAE) technique for detecting and characterizing non-

cooperative spacecraft maneuvers using space-based sensors for spacecraft in close

proximity. A series of classical and variable state multiple model frameworks are im-

plemented, tested, and analyzed through maneuver detection scenarios using relative

spacecraft orbit dynamics. Variable levels of noise, data availability, and target thrust

profiles are used to demonstrate and quantify the performance of the MMAE algo-

rithm using Monte Carlo methods. The current research demonstrates that adaptive

estimation techniques are able to handle unknown changes in the dynamics while

keeping comparable errors with respect to other classical estimation methods.

iv

To those who made the ultimate sacrifice so others could reach for the stars

Ad Astra Per Aspera

v

Acknowledgements

There are many people that I must recognize without whom I believe this journey

through AFIT and beyond would not be possible.

To my research advisor, Joshuah Hess, who has kept me on track and guided me

through my thesis project every step of the way.

To my instructors and advisors at AFIT and USAFA, who inspire me to continue

learning not just throughout my career but throughout my life.

To my parents, Larry and Kristina, who have provided me with every opportunity

to aspire to greatness and a moral foundation to use that greatness for good.

To my sister, Louise, who excites my inner passion for engineering and constantly

reminds me that I could be making more money in the private sector.

To my lovely wife, Felicia, who not only supports me through the toughest times

but is there to pick me up and celebrate the happiest times.

Justin D. Katzovitz

vi

Table of Contents

Page

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Document Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

II. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Statistical Orbit Determination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Applications of Estimation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Least Squares Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.4 The Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.5 Filter Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Multiple Model Adaptive Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Interacting Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Variable State Dimension Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Relative Satellite Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Local-Vertical Local-Horizontal Reference Frame . . . . . . . . . . . . . . 232.3.2 The Hill Clohessy Wiltshire Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Space Sensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Space-based Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.2 Measurement Collection Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.3 Measurement Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

III. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Research Questions Reviewed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Overview of the Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Kalman Filter Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 VSD Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.3 IMM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

3 Scenario Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 Initial Conditions and Noise Factors . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Parameter Study for Maneuver Detection . . . . . . . . . . . . . . . . . . . . 453.3.3 Kalman Filter Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

IV. Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Maneuver Detection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Small Maneuver Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Maneuver Characterization Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4 Dynamic Thrust Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Parameter Study Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 IMM Algorithm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

V. Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Research Questions Answered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Research Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3 Potential Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A. Estimation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B. Additional Scenarios and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C. Nonlinear Dynamical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

viii

List of Figures

Figure Page

1 The Gaussian zero-mean probability density function . . . . . . . . . . . . . . . . . 8

2 VSD filter switching from quiescent model tomaneuvering model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 The LVLH reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Visual diagram of the VSD IMM algorithm . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Visual diagram of the 1G IMM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Position errors for maneuver detection scenario 1 with(a) a single VSD filter and (b) IMM VSD filters; dashedlines indicate the target maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 For maneuver detection scenario 2 (a) the maneuverdetection statistic and (b) the thrust magnitudeestimate vs truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Maneuver detection statistic for small maneuverscenario 1 with (a) a single filter and (b) IMM filters;dashed lines indicate the target maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9 Thrust acceleration estimate for maneuvercharacterization scenario 1 with (a) a single filter and(b) IMM filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

10 Thrust acceleration estimate for maneuvercharacterization scenario 3 with (a) a single filter and(b) IMM filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

11 Thrust acceleration estimate for dynamic thrust scenario . . . . . . . . . . . . . 68

12 Typical model weights for (a) a 1G IMM algorithm and(b) a 2G IMM algorithm; dashed lines indicate thetarget maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13 Model weights for (a) maneuver detection scenario 3and (b) maneuver detection scenario 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

ix

List of Tables

Table Page

1 Critical values of a chi-square distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Common space-based propulsion systems and theirapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Initial parameters for baseline scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Simulation results for baseline scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 The algorithms and filters used for each results section . . . . . . . . . . . . . . . 49

6 Initial parameters for maneuver detection scenario 1 . . . . . . . . . . . . . . . . . 50

7 Simulation results for maneuver detection scenario 1 . . . . . . . . . . . . . . . . . 51



10 Initial parameters for small maneuver scenario 1 . . . . . . . . . . . . . . . . . . . . 56

11 Initial parameters for small maneuver scenario 2 . . . . . . . . . . . . . . . . . . . . 58

12 Simulation results for small maneuver scenario 2 . . . . . . . . . . . . . . . . . . . . 59

13 Simulation results for maneuver characterizationscenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

14 Initial parameters for maneuver characterizationscenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62




18 Initial parameters for dynamic thrust scenario . . . . . . . . . . . . . . . . . . . . . . 67

19 Simulation results for dynamic thrust scenario . . . . . . . . . . . . . . . . . . . . . . 67

x

Table Page





24 Simulation results for small maneuver scenario 1 . . . . . . . . . . . . . . . . . . . . 86



27 Simulation results for dynamic thrust scenario 2 . . . . . . . . . . . . . . . . . . . . 88

28 Simulation results for dynamic thrust scenario 3 . . . . . . . . . . . . . . . . . . . . 88

29 Initial parameters for the relative trajectory scenario . . . . . . . . . . . . . . . . 89

30 Simulation results for the relative trajectory scenario . . . . . . . . . . . . . . . . 90

xi

List of Acronyms

U.S. United States

UN United Nations

USAF U.S. Air Force

DoD U.S. Department of Defense

SSA space situational awareness

JSpOC Joint Space Operations Center

SSN Space Surveillance Network

AFSPC Air Force Space Command

NRC National Research Council

UCT uncorrelated target

RPO rendezvous and proximity operations

GEO geosynchronous orbit

SOD statistical orbit determination

MMSE minimum mean square error

LEO low Earth orbit

BLS batch least squares

KF Kalman filter

EKF extended Kalman filter

xii

MMAE multiple model adaptive estimation

IMM interacting multiple models

UKF unscented Kalman filter

UT unscented transformation

FES fixed epoch smoothers

IMM interacting multiple models

PDF probability density function

VSD variable state dimension

MD Mahalanobis distance

ED Euclidean distance

NMC natural motion circumnavigation

ROEs relative orbital elements

EO electro-optical

LIDAR light detection and ranging

IOD initial orbit determination

1G first generation

2G second generation

xiii



I. Introduction

1.1 Motivation

Since launching the first satellites into Earth orbit, the United States (U.S.) and

its military have treated the space domain as the ultimate high ground [1]. The

space-based technology requirements of today’s world have created an arena of

contest, congestion, and competition among other space-faring nations [2].

Approximately 60 nations as well as a multitude of commercial and academic

satellite operators currently work with thousands of space assets in Earth orbits [3].

The 1974 United Nations (UN) Convention on Registration requires nations

launching objects into space to register basic orbit parameters and general

spacecraft function [4]. Despite providing a symbolic step forward in international

cooperation with regards to the space domain, the required basic orbit parameters

provide little information necessary for accurate and real-time orbit determination

for the growing list of active spacecraft in Earth orbits. In the years since the UN

convention, the U.S. Air Force (USAF) has conducted its own mission to provide

precise tracking of objects in space for its own assets as well as any other man-made

object large enough to track [2].

The U.S. Department of Defense (DoD) and the USAF emphasize continued

research of space situational awareness (SSA) to protect the US and allied space

capabilities from an increasingly congested space environment [3]. From the Joint

1

Space Operations Center (JSpOC) and the dedicated set of ground radar and

electro-optical (EO) sites around the globe that make up the Space Surveillance

Network (SSN), the DoD actively tracks approximately 22,000 objects in Earth

orbits [2]. Current research throughout the USAF focuses on improving the precise

predictions of man-made spacecraft and debris with the primary objective of

collision avoidance [5], but extended efforts must be taken to perform accurate orbit

determination of additional objects in space due to the high level of uncertainty in

the space environment. JSpOC requires constant improvement in SSA algorithms in

order to support the U.S.’s increasingly important mission of protecting its space

assets [6].

National security concerns limit international cooperation with respect to the

current orbits and future maneuver plans of space assets [7]. Without the open

source sharing of precise orbital information or plans to maneuver space assets, the

U.S. and other space-faring nations face an increased risk for spacecraft collisions.

As seen with the 2009 collision of the commercial communications satellite Iridium

33 and the decommissioned Russian military communications satellite Cosmos 2251,

collisions in space can easily turn two highly capable assets into thousands of pieces

of orbital debris. JSpOC uses a collection of astrodynamic algorithms standardized

by Air Force Space Command (AFSPC) and assessed by the National Research

Council (NRC) to track and estimate the location of objects in space and provide

collision warning to any spacecraft at risk [2].

Much of the SSA effort is used to associate ground-based sensor readings with

specific items in the space catalog [2]. When a spacecraft performs an unannounced

maneuver, the errors in sensor measurements become greater than the required

confidence interval of the cataloged object and its original orbit, creating an

uncorrelated target (UCT) in the JSpOC database [2]. More research is needed to

2

improve tracking, prediction, and estimation of non-cooperative spacecraft that

perform unknown maneuvers at unknown times to increase target correlation and

the overall SSA mission [3].

1.2 Problem

This study assesses the detection and characterization of non-cooperative

spacecraft maneuvers with space-based sensors in rendezvous and proximity

operations (RPO) using adaptive estimation techniques. Previous research by Goff

et al. demonstrated the ability to detect and track unknown maneuvers of

non-cooperative spacecraft using ground-based radars [7]. Although adaptive

estimation techniques have claimed to be more effective at detecting and tracking

unknown maneuvers than traditional orbital estimation techniques [8], Goff admits

that future work is necessary to evaluate scenarios where a spacecraft maneuvers

into an area not covered by ground-based radars. The current research applies an

estimation architecture similar to that in [8] using space-based sensors for spacecraft

in geosynchronous orbit (GEO), filling in the limitations of ground-based sensors.

The motivation of this research identifies three potential interests for the

operational use of adaptive estimation techniques for maneuver detection and

characterization. Serving as the single most up-to-date space object tracking

system, JSpOC needs an accurate and robust orbit determination network.

Consistent detection and tracking of orbital maneuvers will improve JSpOC’s ability

to provide early warning prediction of satellite collisions. Detecting maneuvers of

non-cooperative spacecraft will also provide insight to the commercial space

industry, improving the state-of-the-art in orbital estimation for on-orbit rendezvous

and servicing missions. This research also provides an assessment on applications of

space-based assets to the maneuver detection problem, which will provide valuable

3

information to the USAF space acquisition community when applying adaptive

estimation algorithms to future operational space missions. The overall research

objectives of this study are tailored with these interests in mind to provide a clear

path forward in terms of future operational applications.

The study is divided into three research questions focused on examining current

adaptive estimation techniques and applications to relative satellite motion and

proximity operations as well as assessing the performance of an adaptive estimation

algorithm through a realistic parameter study. The following research questions

shall be answered by the conclusion of this study:

• Can adaptive estimation techniques be applied to detect and characterize

non-cooperative spacecraft maneuvers in satellite close proximity operations?

• How do sensor source and type, data rate, maneuver magnitudes, and relative

trajectory affect the performance of an adaptive estimation algorithm?

• For what types of scenarios does an adaptive estimation algorithm fail to

detect a maneuver or fail to characterize an accurate maneuver magnitude?

1.3 Document Overview

This document consists of five chapters, the first of which is an introduction to

the motivation behind this study and its research objectives. Chapter 2 is a

comprehensive literature review on the background, theory, and methodology

behind this research, to include previous research done on statistical orbit

determination (SOD), adaptive estimation, relative satellite motion, and space-based

sensor analysis. Chapter 3 describes the methodology used in this study, presenting

all of the necessary algorithms and equations discussed in Chapter 2 that will form

the foundation for the variety of maneuver detection and characterization scenarios

4

necessary to accurately validate the different adaptive estimation algorithms used in

this study. Chapter 4 contains the results and analysis of each maneuver detection

and characterization scenario along with discussions assessing efficacy, timeliness of

convergence, and comparisons to traditional estimation algorithms. This document

concludes with Chapter 5, which summarizes the study and emphasizes the

significance of the research while providing recommendations for future work.

5

II. Literature Review

The task of detecting maneuvers of non-cooperative spacecraft using adaptive

estimation for RPO scenarios requires an extensive research effort in the areas of

orbit determination, estimation theory, and relative satellite motion. This chapter

reviews relevant literature that provides the groundwork for each topic of interest,

including the derivations of notable equations and algorithms where necessary. This

chapter also emphasizes previous research done in the area of spacecraft maneuver

detection, providing realistic applications of current research into the overall topic of

orbital estimation and tracking.

2.1 Statistical Orbit Determination

SOD is the generalized term used to describe the application of estimation

theory and how to account for errors in observation measurements and uncertainties

in the dynamics for orbit determination. SOD utilizes the principles of estimation

theory through minimizing residual errors while predicting a spacecraft’s orbital

trajectory, often referred to as the states of the system. The principle of a minimum

mean square error (MMSE) estimation algorithm is to take a series of dependent

variables or measurements and produce an approximation of the state variables as

well as a mean squared estimate for the error in each state. In satellite dynamics,

the mean square error estimate produced is represented by the covariance matrix,

which Wiesel defines as the correlation of the error estimate between each state

variable [9]. The following sections supply the theory and basic derivations of each

estimation technique used in this study and their current applications to SOD and

the overall USAF SSA mission.

6

2.1.1 Applications of Estimation Theory

Originally developed by Carl Friedrich Gauss in the early 1800s, estimation

theory uses statistics to estimate the error in a measured variable and correlate the

estimate to a confidence factor for the measured variable. Gauss revolutionized the

field of orbit determination by focusing on minimizing measurement and calculation

errors instead of attempting to find the perfect dynamical equation to represent

orbital motion [10]. Over a century later, the advancement of technology brought

forth new estimation algorithms that take advantage of increasing computational

speed and accuracy. Today, the USAF utilizes modern estimation theory to

minimize observation errors when tracking space objects [2]. Tapley et al. provides

an in-depth analysis at the estimation theory behind SOD and presents a realistic

approach to building an orbit determination problem through defined orbital

dynamics and initializing error and covariance estimates [11]. Before discussing the

theory behind the estimation techniques used in this study, the estimation problem

is defined in terms of the inherent error in a dynamical system as well as the states

that define the system.

Gauss made the assumption that there is no perfect equation to describe the

motion of a system, establishing the need for deterministic estimation theory.

Modern estimation theory makes a second assumption in that there are no perfect

measurements to determine the states of a system, requiring the need for stochastic

estimation theory [9]. Through his founding of probability theory, Gauss defined

imperfections in dynamical systems as random processes, otherwise known as noise

[10]. For most dynamical systems, the noise in a state estimate can be defined by a

normally distributed probability density function, also known as the Gaussian

distribution as shown in Figure 1. Although noise is defined to be a random process,

the assumption that the noise of the error is normally distributed is not only proven

7

by the central limit theorem but is also observed in the real world all the time. A

complete proof of the central limit theorem can be found in [9].

Figure 1. The Gaussian zero-mean probability density function

A state estimate described by Gaussian white noise is fully defined by a mean of

the state, E(x), and a standard deviation of the state, σx. A dataset defined by a

Gaussian distribution implies that approximately 99% of the points in the dataset

are within 3σ of the mean of the set. Using Figure 1 as an example, E(x) = 0 and

σx = 1. Unless otherwise stated, all instances of error in the dynamical and

measurement equations of this study are assumed to be zero-mean, independent,

white Gaussian noise. These assumptions for the current study imply that every

state estimate has a normally distributed error centered about its mean, and each

state error is uncorrelated with any other state error. Noise and errors in the system

are further defined for this study in Chapter 3.

General applications of estimation theory for dynamical systems are built to

estimate states that are related in some way to the variable dynamics of the system,

8

represented by the equations of motion. Wiesel defines a general state vector x,

which defines the states of the system [9]. The system is defined in Equation (2.1)

by a series of differential equations that relate the changes of the state vector over

time, denoted by the generalized function f , and the confidence in the dynamical

model, denoted by the random process w(t)

dx

dt= f (x, t) + w(t). (2.1)

Another way to represent the dynamics is through a state transition matrix.

The state transition matrix relates a previous state vector to a new state vector

separated by some dependent variable, in this case a discrete time step (t2 - t1).

Shown in Equation (2.2), Stengel and other textbooks on estimation theory use the

state transition matrix to define a discrete-time system with process noise [11, 12]

x(t2) = Φ(t2, t1)x(t1) + w(t1) (2.2)

where Φ(t2, t1) is defined as the state transition matrix, which for nonlinear systems

can vary over each update of the state vector. The term for process noise, w(t1), is

considered a disturbance input for the system, and is still assumed to be white

Gaussian noise.

Estimation theory also defines the relationship between the measurements and

the state vector, along with the uncertainties associated with the measurements

[12]. The measurement vector, z(t), is defined in Equation (2.3)

z(t) = y(t) + v(t) = h[x(t)] + v(t). (2.3)

Stengel uses the output vector, y(t), to relate the state vector, x(t), to the

measurement vector, z(t) with the measurement error represented by a noise

9

component, v(t) [12]. The output vector can be any combination of the states that

the user has interest in, which is defined by the measurement basis function, h[x(t)]

[13].

In most cases, it is simple and efficient to define the state vector x as the

position and velocity vector when dealing with particle dynamics. However, it can

be useful and at times necessary to include other pieces of information regarding the

dynamics of the system in the state vector such as coordinate transformations or

osculating orbital elements. As long as the equations of motion can be defined with

respect to the state vector, estimation theory can be used to track the changes to

the state vector and the errors in the estimate of each state in the system.

2.1.2 Least Squares Estimation

Least squares estimation can be traced back to the original foundations of

estimation theory [10]. By using the probability assumptions described in section

2.1.1, Gauss invented the least squares method to obtain orbits on objects with a

limited amount of observations [9]. The main principle of deterministic least squares

is to calculate the estimate state, x̂, such that the square of the residuals is

minimized. The calculation of the residuals, r̄, shown in Equation (2.4), is usually

defined as the difference between the observed measurements and the predicted

measurements, described earlier in Section 2.1 as the measurement vector and a

function of the state vector [14]. A full derivation of both the linear and nonlinear

least squares algorithm can be found in Vallado’s text [15].

r̄ = z(t)− ŷ(t) = z(t)− h[x̂(t)] (2.4)

Although the least squares algorithm is effective in minimizing the residual

error of the system, problems arise with SOD applications in terms of both

10

computational time and accuracy. Every observation available is incorporated into

the least squares algorithm, and the state vector is sized on the order of the number

of measurements. When these measurements are limited, as in the case of Gauss

estimating the orbit of the asteroid Ceres [9], the least squares solution computed

by hand is precise. Over the course of observing a satellite in low Earth

orbit (LEO), however, an increasing number of measurements over time causes the

memory requirements on a program to become significant.

Since the dynamics of an orbit are inherently nonlinear, the forces affecting the

spacecraft at one observation could differ greatly from the forces at another

observation over the course of many orbits around the Earth. This causes problems

in the least squares solution, as the dynamical state model cannot be easily changed

in real-time [14]. As the external forces on a satellite in orbit, such as aerodrag or

the effects of J2, grow increasingly nonlinear, treating older measurements with the

same confidence as newer observations will cause significant errors in the least

square solution [14].

One of the ways to solve the computational time and accuracy issues with the

least squares algorithm is through batch processing. The main idea behind the

batch least squares (BLS) algorithm is to continuously solve the least squares

problem with available data without having to continuously calculate old data.

Vallado provides the framework for solving the least squares problem through batch

processing [15], while Tapley et al. steps through the derivation and application of

the BLS algorithm [11]. This still produces a linearized solution to the nonlinear

problem, so iterating may not converge on a minimized residual solution [15].

The BLS algorithm is a widely used technique for SOD because it is designed to

handle a small number of observations over a significant period of time while still

minimizing the error between the measurements and the state estimate. Since the

11

SSN ground radar system is tasked with tracking tens of thousands of space objects

daily, the JSpOC orbit tracking and SSA missions are a useful application of the

BLS algorithm [2]. Batch processing is used in these instances of SOD, which can

take a series of observations over significant periods of time and fit the data to the

predicted orbit while minimizing the residuals. Although the BLS technique is

useful for a multitude of ground measurements spread out over many orbital

periods, problems arise when system perturbations are not entirely known and are

therefore modeled incompletely [16]. Due to the real-time requirements inherent in

the maneuver detection problem analyzed in the current study, more sequential

estimation methods are required over the BLS algorithm.

2.1.3 Kalman Filtering

A major issue with the least squares estimation technique is that the converged

state and covariance matrix are based on a large batch of data that may have

accumulated error over a prolonged period of time. No matter how accurate the

estimate is, the least squares algorithm is always based on an epoch time and may

not have a precise estimate for any state at a future epoch time. The solution to

these inherent problems is sequential estimation, or computing the best state

estimate of a time-varying process [15]. The Kalman filter (KF) solves the same

least squares problem as BLS, but tries to minimize errors through sequential

estimation [17].

Vallado cites two major differences between the least squares technique and the

KF [15]. First, the KF continuously updates the epoch time, only predicting the

state estimate at a future observation time. Second, the KF keeps all past

information in the current estimate and covariance matrix, eliminating the need to

continuously recalculate any past states or measurements at each time step.

12

The dynamical equations of motion as well as the measurement equations for

orbit mechanics are often nonlinear in nature, and the linear KF is not always

applicable. Therefore, the extended Kalman filter (EKF) is sometimes necessary

[15]. Wright explains that the EKF is more dynamic than the BLS and is more ideal

for orbit determination [18]. The EKF has a similar algorithm as the linear KF, but

uses a Taylor series approximation at each time step in the propagation to

temporarily linearize the dynamics and apply the optimal least squares solution.

Derivations of the linear KF and EKF can be found in many astrodynamics and

estimation textbooks [9, 11, 12] as well as other works utilizing sequential

estimation [13, 19]. Both the KF and the EKF start with an initial guess of the

state and covariance matrix

x̂(t0) = x̂0

P(t0) = P0.

(2.5)

Using Equations (2.2) and (2.3) to define the state and measurement equations,

the process and measurement noise are assumed to be independent, zero mean,

normal probability distributions, with covariances shown in Equation (2.6)

p(w) ∼ N(0,Q)

p(v) ∼ N(0,R)(2.6)

where the process noise covariance , Q, and the measurement noise covariance, R,

are free to change at each time step to represent the confidence in the dynamics or

the data at any time during the propagation of the filter.

13

The first step in the KF is to propagate the state vector from xk−1 to xk

depending on the dynamical equation of the system. The covariance matrix is also

propagated from Pk−1 to Pk based on the mean squared error of the predicted

state. The predicted state, xk, is then corrected by a combination of the

measurement basis function, hk[x−k ], the predicted state covariance matrix, P

−k , and

the measurement covariance matrix, Rk. The correction calculation shown in

equation 2.7 is called the Kalman gain. In both the KF and EKF algorithms, the

Kalman gain is calculated by the same equation

Kk = P−k H

Tk (HkP

−k H

Tk + Rk)

−1 (2.7)

where the matrix Hk is a linear measurement basis function multiplied by x̂−k to get

the output vector, y. In Equation (2.7), the − superscript represents the previous

predicted estimate in the algorithm, while the updated state and covariance

estimate shown in Equation (2.8) is represented by the + superscript. The state and

covariance estimate are updated using the same equations for the linear and

nonlinear case.

x̂+k = x̂−k + Kk(zk −Hkx̂

−k ) (2.8a)

P+k = (I−Kkhk[x̂−k ])P

−k (2.8b)

Notice that the last term in Equation (2.8a) is the residual vector defined in

Equation (2.4), only this time it only represents the residual vector at the specific

time step instead of at every measurement for the least squares algorithm.

Looking back at Equations (2.5) to (2.8b), it would seem that the linear KF

and the EKF follow the exact same algorithm, which during the

14

prediction-correction stage of the algorithm is completely true. The primary

difference between the KF and the EKF is that the EKF must linearize its

dynamics and measurement basis function in order to propagate in discrete time. If

the dynamics follow the generalized functions shown in Equation (2.1) and (2.3),

then the state estimate is propagated by Equation (2.9), and the linearized

measurement basis function is shown in Equation (2.10).

xk =

∫ tktk−1

[ẋk−1]dx+ xk−1 (2.9)

Hk =∂z

∂x−k(2.10)

Using a linearized form of the dynamics and the measurements at each time

step, the EKF can easily transform the nonlinear dynamic systems seen in orbit

mechanics into a simplified model that can be propagated through the KF

algorithm. However, no matter how small the time step, any linearized model of

nonlinear dynamics are bound to have inherent errors, which can be unacceptable

for SOD.

2.1.4 The Unscented Kalman Filter

Accuracy concerns arise with the EKF because the algorithm must differentiate

the measurement and dynamic functions with respect to the state vector in order to

linearize the problem at every time step. Julier and Uhlmann developed a nonlinear

Kalman filter called the unscented Kalman filter (UKF) for improved performance

and accuracy [20]. While the EKF simply linearizes the dynamics of the system

through each step in the estimation, the UKF overcomes the difficulties that arise

from linearization by propagating the mean and covariance of the state vector

15

through nonlinear transformations [21] . Instead of relying on numerical integration

or linearized propagation, the UKF moves a weighted distribution of critical points

based on the current estimate of the covariance matrix. The weighting methods

used in this study for the nonlinear unscented transformation (UT) are outlined in

Section 3.2.1.

As explained in Teixeira et al., the UKF is similar in computational efficiency

and superior in accuracy to the EKF when implemented properly for orbit

determination [22]. In addition, Teixeira et al. conclude that the UKF is able to

converge uniformly better than the EKF for time-sparse measurements [22]. Pardal

et al. also show that the UKF outperforms other filters when observations are less

frequent, specifically testing this hypothesis with pseudo range observations [23].

2.1.5 Filter Smoothers

At the end of each scenario, a backwards smoother applied to the filter can

further improve the estimated state. Wright and Woodburn explore different

combinations of fixed epoch smoothers (FES) with an EKF to improve state

estimates [24]. The FES is ideal for discrete satellite observations used in orbit

estimation; new measurements processed through the EKF in real-time are also

used to recursively update the estimated state and covariance at some previous

epoch time. Helmick et al. examines a fixed-interval smoothing algorithm using an

adaptive estimation framework, which updates all estimates within an interval from

the current estimate [25]. This approach can be costly for orbit estimation in

real-time because the algorithm requires n2 smoothers working in parallel for every

n models.

Another type of smoother developed for the UKF by Särkkä is the unscented

Rauch-Tung-Striebel smoother [26]. Instead of combining the results of a forward

16

running UKF for the backward-working algorithm, a backward smoother is used to

calculate suitable corrections to the forward algorithm. This concept can improve

the state error for scenarios with large nonlinearities and unknown dynamics such as

a maneuver because it can update previously erroneous state estimates with newly

propagated dynamics. However, because these smoothing algorithms require varying

levels of post-processing, they are not ideal for real-time estimation scenarios and

will not be tested in this study.

2.2 Multiple Model Adaptive Estimation

The NRC has expressed the need for considering future research in multiple

model adaptive estimation (MMAE) with regards to the SSA mission and more

specifically maneuver detection [2]. Tracking a non-cooperative spacecraft and

detecting unknown maneuvers requires an adaptive estimation technique because

the noise components of the non-cooperative spacecraft are unknown and must be

estimated. Magill outlines the fundamentals of adaptive estimation and its

applicable derivations [27]. MMAE uses a bank of KFs that all make different

assumptions about the dynamics of the system, specifically in the covariance of the

process noise, Q.

The adaptive framework allows for different ways to account for unexpected

changes in the state or covariance estimates. Each of the filters has a series of

weighting coefficients that can change based on algorithm specific rules regarding

calculations and filter initializations. Li et al. presents an adaptive filter using a

series of UKFs to approximate and adjust the process noise covariance throughout

the propagation of the algorithm [28]. Moose developed an adaptive state estimator

for the general maneuvering target problem and concluded that the adaptive

estimator with a band of KFs is certainly superior to the linear KF [29]. Although

17

proven accurate in estimating uncertainties in the dynamics of the target, the

adaptive estimation technique still must make a series initial guesses on the process

noise covariance matrix. Without interaction, the MMAE could still diverge if the

guesses of the process noise covariances are not accurate.

2.2.1 Interacting Multiple Models

The concept of interacting multiple models (IMM) is used to solve the problem

of poor initial guesses in the adaptive estimation algorithm. Li et al. states that the

IMM method is the prevailing approach to modern maneuvering target tracking

[30]. The IMM combines the inputs of several models at each time step and uses the

statistics of the residuals to weigh the impact of each model at each step. Compared

to the general adaptive estimation method, the IMM algorithm creates a probability

density function (PDF) of the model weights, which produces the converged results

as a combination of multiple models that could be the correct covariance estimate.

The specific IMM algorithm and PDF used in this study are outlined in Section

3.2.3.

When applying general MMAE techniques to filter through unknown

maneuvers, covariance inflation must occur to prevent divergence [31]. Covariance

inflation is the process of assuming no confidence in the dynamical model during the

maneuver so that the measurement basis function can dominate the state estimate

[11]. Through covariance inflation, however, a trade off occurs. A large covariance

causes a high probability of convergence, but also a high chance for errors in the

state estimate. The IMM filter prevents the need for determining optimal covariance

sizes that are only valid for specific dynamical systems by providing a method to

mix different covariance estimates and weighting the results based on the likelihood

probability calculated for each model [32]. For this reason, the IMM method is

18

considered suboptimal because it may not converge on the exact covariance for the

target orbit during maneuver detection, but as Goff shows in [7], the IMM method

is effective in maneuver detection algorithms when limited information is known

about non-cooperative spacecraft maneuvers.

The IMM framework steps closer towards being able to handle the maneuver

detection problem in that it accounts for the uncertainty in the dynamics of the

non-cooperative spacecraft unknown to the observer through adaptive covariance

analysis. However, large changes in the dynamics, such as an unknown active thrust

maneuver at an unknown time, cannot be accounted for simply through covariance

inflation and adaptive estimation.

2.2.2 Variable State Dimension Filter

The variable state dimension (VSD) filter is a solution to handling major

unknown changes in the dynamics of a maneuvering spacecraft. Bar-Shalom et al.

was the first to apply a variable dimension filter to a general maneuver tracking

scenario [33]. The VSD filter is optimal for dealing with unknown maneuvers

because the filter has the ability to add or subtract states based on the deviation of

the estimated state from the measured state, defined earlier as the residual vector.

High residuals are potentially caused by a fundamental error in the equations of

motion of the system, but a VSD that monitors residuals could account for errors by

adding additional states to the state vector (such as a thrust vector) for a spacecraft

changing its dynamics by maneuvering.

Bar-Shalom et al. expand the algorithm of the VSD filter in their text [34]. The

fundamentals of the VSD filter can be structured as any KF previously discussed in

Section 2.1.3. In this case, two filter models are used: the quiescent model, with

states defined as the position and velocity vector of the target, and the maneuvering

19

model, which tracks an acceleration vector as additional states to the system. The

model switching indicator uses a fading memory average of the sequentially

calculated residuals of the filter based on the quiescent model, seen in Equation

(2.11)

Ψk = r̄TS−1k r̄ (2.11)

where Sk is the covariance matrix of the residual vector, also known as the

measurement covariance. The scalar value defined in Equation (2.11) is known as

the Mahalanobis distance (MD), and is referred to as the maneuver detection

statistic in the current study. Compared to the commonly used Euclidean

distance (ED), the MD takes into account the correlation in the data through the

measurement covariance matrix [35], which allows the estimation algorithm to

converge on a minimized set of residuals without detecting a false maneuver.

The MD value is based on a chi-square distribution, and the maneuver

threshold used to detect the maneuver start and stop times is based on a two-sided

test at a significant level α = 0.0005, which corresponds to a probability of p =

0.999 that the maneuver detection statistic is less than the critical value. Equation

(2.12) shows the chi-square distribution PDF:

PDF(x; k) =xk/2−1e− x/2

2k/2Γ(k2

) . (2.12)Here x is the maneuver detection statistic, k is the degrees of freedom for the

system, and Γ denotes the gamma function. Table 1 shows the critical values of a

chi-square distribution based on the degrees of freedom for the system. Based on

the critical values for the 4 and 6 degrees of freedom for the states of the system

20

defined in this study, the maneuver threshold is maintained at a constant value of

20 throughout the maneuver detection scenarios.

Table 1. Critical values of a chi-square distribution

DOF Critical Value for p = 0.999

1 10.828

2 13.816

3 16.266

4 18.467

5 20.515

6 22.458

Because the change in the residual vector over the observation time varies

depending on the dynamics of the target, the VSD algorithm assumes that the

maneuver started before the VSD detects the maneuver. Bar-Shalom et al. define

the effective window length of detecting a maneuver as the multiplicative sum of a

weighting value, α, as the discrete time counter, k, approaches infinity [34]. Since

the weighting value is defined between 0 < α < 1, the effective window length, ∆α,

for detecting a maneuver is described in Equation (2.13).

∆α =1

1− α(2.13)

Although Bar-Shalom concludes that this metric is an acceptable window to

backtrack through the data and estimate the start of the maneuver, there is no

metric in defining the weighting matrix α besides the intuitive confidence in the

past measurements. A low α constitutes a low confidence in the previous residual

vectors in the filter, and Equation (2.13) therefore assumes the maneuver started

earlier than if there was a higher confidence in the previous estimates. Figure 2

21

shows the transition of estimating using the quiescent model to estimating using the

maneuvering model using the detection criteria outlined by Bar-Shalom et al. [33].

Although not explicitly stated, the metric for detecting the end of the maneuver is

also calculated using the same algorithm that detects the start of the maneuver.

Figure 2. VSD filter switching from quiescent model to maneuvering model

Using the extensive review of estimation theory shown in Section 2.1, this study

uses the background shown in Section 2.2 to develop the algorithm necessary to

detect unknown maneuvers of non-cooperative spacecraft using an observer satellite

collecting measurements in close proximity. Specifics regarding the adaptive

estimation techniques used in the current study are presented in Chapter 3. The

22

following sections review the necessary background, theory, and equations necessary

to describe the motion of spacecraft flying relative to each other in Earth orbits.

2.3 Relative Satellite Motion

As stated in Section 1.2, this study applies adaptive estimation techniques

discussed in Section 2.2 to RPO scenarios with the objective of non-cooperative

maneuver detection and characterization. The differential equations established in

Equation (2.1) for any estimation algorithm must be able to accurately describe the

dynamics of satellites relative to each other. Relative satellite motion provides a

convenient and efficient method to define the dynamics of RPO spacecraft without

the need for an inertial reference frame.

For every scenario described in this study, consider two satellites in Earth

orbits, one identified as the “chief” and the other identified as the “deputy”. For the

purposes of consistency, the chief in this study is also considered the “observer”, or

the spacecraft taking measurements on the deputy, which will be considered the

“target”. When deriving the equations of motion for RPO scenarios, a new

non-inertial coordinate frame must also be considered to describe the orbit of the

target with respect to the observer.

2.3.1 Local-Vertical Local-Horizontal Reference Frame

As with all dynamical systems, the reference frame used to describe satellite

motion is just as important as the equations of motion themselves. The reference

frame most often used to describe relative satellite motion is the local-vertical

local-horizontal (LVLH) frame, also called the Hill frame for his original derivation

in describing the Moon’s orbit around Earth with respect to the Sun [36]. The

origin of the LVLH frame is centered at the chief, with the x-axis pointing in the

23

direction of the chief’s position vector with respect to the Earth, the z-axis pointing

normal to the orbital plane, and the y-axis completing the right handed coordinate

system. The y-axis is generally pointed in the along-track direction, and if the

chief’s orbit is circular then the y-axis is directly aligned with the chief’s velocity

vector. Figure 3 shows the LVLH reference frame centered at the chief spacecraft,

providing a visual relationship between the states [x,y,ẋ,ẏ] and the space-based

sensor measurements range (ρ), range rate (ρ̇), and azimuth (α). Further discussion

regarding space-based measurements can be found in Section 2.4.1.

Figure 3. The LVLH reference frame

2.3.2 The Hill Clohessy Wiltshire Model

The Hill Clohessy Wiltshire (HCW) model is the most widely used set of

equations that accurately describe the relative motion of two spacecraft operating in

close proximity. The equations of motion were originally developed by Clohessy and

Wiltshire in 1960 for satellite rendezvous [37] and are similar to Hill’s equations of

24

motion in his lunar theory [36]. Derivations of the HCW equations can be found in

many astrodynamic textbooks [16, 38, 39]. The full nonlinear equations of relative

motion are shown in Equation (2.14)

ẍ− 2nẏ − ṅy − n2x− µr2

=−µr3d

(r + x) + fx (2.14a)

ÿ + 2nẋ+ ṅx− n2y = −µr3dy + fy (2.14b)

z̈ =−µr3dz + fz (2.14c)

where x, y, and z are the position components; ẋ and ẏ are the velocity components;

and ẍ, ÿ, and z̈ are the acceleration components of the target in the LVLH frame.

The variable r and rd refer to the distance of the chief and deputy with respect to

the Earth, the variables n and ṅ refer to the mean motion of the chief and its first

time derivative, and the variable µ refers to the gravitational constant of the Earth.

The full nonlinear equations of motion are difficult and time consuming to

propagate, and the estimation algorithms presented in Section 2.1 are designed to

handle small unknown errors in the dynamics, which allows for some simplifying

assumptions. The three major assumptions that allow for the full linearized model

of the HCW equations are that the two spacecraft are in Keplerian motion, so the

only force modeled is Earth’s gravitational field as a point mass; the chief is in a

circular orbit, so its mean motion is assumed constant; and the distance between

the satellites is small compared to their orbital radii, so rd ≈ r. For many formation

flying missions in a near-circular orbit and with proper estimation techniques, these

assumptions tend to be valid [13, 40]. The simplified HCW equations are shown in

Equation (2.15):

25

ẍ− 2nẏ − 3n2x = fx (2.15a)

ÿ + 2nẋ = fy (2.15b)

z̈ + n2z = fz. (2.15c)

These equations are written in the LVLH reference frame, where x, y, and z are

used to describe the deputy’s position with respect to the chief in the radial,

along-track, and cross-track directions, respectively. The term n in Equation (2.15)

is used to denote the mean motion of the chief, which can be found using the

Earth’s gravitational constant, µ, and the semi-major axis of the chief, ac, as seen in

Equation (2.16)

n =

√µ

a3c. (2.16)

In this study, another simplified nonlinear form of the HCW equations is

explored. The assumptions of Keplerian motion and a circular chief are still valid,

but removing the relative distance assumption allows for scenarios with significant

distances between the chief and the deputy without losing accuracy. The nonlinear

HCW equations without the relative distance assumption are shown in Equation

(2.17).

ẍ− 2nẏ − n2x− µr2

=−µr3d

(r + x) + fx (2.17a)

ÿ + 2nẋ− n2y = −µr3dy + fy (2.17b)

z̈ =−µr3dz + fz (2.17c)

26

The right-hand side of Equations (2.14), (2.15), and (2.17) allow for any

external forces acting on the system to be added to the dynamics as a perceived

relative acceleration on the system, which is used in this study as the acceleration

force vector of the deputy when conducting maneuvers.

Assuming the linearized HCW equations of motion shown in Equation (2.15)

have no external forces acting on the system, the analytical solution to the HCW

equations is shown in Equation (2.18)

x =ẋ0n

sin(nt)− (3x0 +ẏ0n

) cos(nt) + (4x0 +2ẏ0n

) cos(nt) + (4x0 +2ẏ0n

) (2.18a)

y =2ẋ0n

cos(nt) + (6x0 +4ẏ0n

) sin(nt)− (6nx0 + 3ẏ0)t−2ẋ0n

+ y0 (2.18b)

z =ż0n

sin(nt) + z0 cos(nt) (2.18c)

ẋ = ẋ0 cos(nt) + (3nx0 + 2ẏ0) sin(nt) (2.18d)

ẏ = −2ẋ0 sin(nt) + (6nx0 + 4ẏ0) cos(nt)− (6nx0 + 3ẏ0) (2.18e)

ż = ż0 cos(nt)− nz0 sin(nt) (2.18f)

where x0, y0, etc. are the relative initial conditions of the deputy at some epoch

time, t0. From Equation (2.18e), HCW dynamical system experiences simple

harmonic motion when the following constraint is met:

ẏ0 = −2nx0 (2.19)

Using the constraint from Equation 2.19, the deputy spacecraft follows a

stabilized 2x1 elliptical trajectory relative to the chief, referred to in other RPO

works as natural motion circumnavigation (NMC) [41]. An NMC trajectory is a

27

convenient relative orbit that is used to create initial conditions for the scenarios

presented in this research.

Using the equations of motion outlined in this section, the estimation algorithm

developed in Chapter 3 is able to relate the current states of the system to the

future states of the system in a way that is realistic to the relative satellite motion

aspect of the study. Although there are major assumptions made to simplify the

relative satellite equations of motion, using the process noise covariance analysis

discussed in Section 2.2 can handle errors in the dynamics of the system while still

using the simplifying assumptions in the algorithm. The final equations that are

necessary to the estimation algorithm relate the measurements collected by the

observer on the target to the relative orbital states of the target.

2.4 Space Sensor Analysis

Although the relative satellite motion dynamics defined in Section 2.3 use the

coordinates x, y, and z in the LVLH frame to derive the equations of motion, the

sensor measurements that are fed into the estimation algorithm do not directly

measure position and velocity in LVLH frame coordinates. As mentioned in Section

1.2, this study analyzes an adaptive estimation algorithm against multiple types of

measurements and measurement noise levels to make indications between algorithm

performance and the quality of space-based sensors required. The types of

measurements analyzed in this study are a combination of range-azimuth-elevation

and range-range rate measurements.

2.4.1 Space-based Measurements

Escobal details several techniques for orbit determination using combinations of

available data [42]. A common set of raw measurements obtained from a

28

space-based sensor is range, azimuth, and elevation, defined similarly to ground

based measurements but here in the LVLH frame [13]:

z = h[x(t)] =

ρ

α

�

LV LH

=

√x2 + y2 + z2

arctan yx

arcsin zρ.

(2.20)

Equation (2.20) is used as the nonlinear measurement basis function to relate the

measurements back to the states of the problem. Another type of space-based

sensor capability to be analyzed in this study is collecting measurements of range

and range rate. This allows us to have some insight into not only the position of the

deputy but also its velocity relative to the chief. The range and range-rate nonlinear

measurement basis function is presented in Equation (2.21) [9]:

z = h[x(t)] =

ρρ̇

=√x2 + y2 + z2

xẋ+yẏ+zż√x2+y2+z2

. (2.21)2.4.2 Measurement Collection Techniques

A significant consideration when running estimation algorithms using

space-based measurements outlined in Section 2.4.1 is the frequency of available and

accurate sensor data. Much of this study assumes that sensor data is widely

available on the observer looking at the target, but a space-based environment

breeds a multitude of opportunities for large data errors, especially in

non-cooperative scenarios.

Much of the previous literature on space-based observability analysis focuses on

cooperative measurement collection for formation flying, but many measurement

collection techniques can be applied to a non-cooperative scenario without any

major fundamental technology upgrades. One of the most common approaches to

29

space-based SSA is the angles-only approach because it is useful at gathering

measurements using solely vision-based navigation; Gaias discusses this approach in

a non-cooperative setting using only a space-based camera [43]. Gaias enhances the

accuracy of space-based servicing missions by converting angles-only data into

relative orbital elements (ROEs). However, acquiring range data can be crucial

when dealing with a target in close proximity to the observer, which requires more

complex measurement collection techniques.

Junkins discusses vision-based navigation using a position sensing diode for

RPO [44], but this requires targeting beacons to sense certain wavelengths of light,

which is impossible in the non-cooperative scenario. Whittaker shows that

space-based measurements can be acquired using a photometric sensor to correlate

light intensity with range from the target [45]. Although this technique improves

upon an angles-only approach, it requires reflected light from the Sun as well as

sensor calibrations based on the material properties from the target spacecraft,

which may not be available in a non-cooperative scenario. Krutz analyzes a

radiometric sensor in a space-based mission to detect and track spacecraft debris

[46]. Although this is an ideal non-cooperative scenario, Krutz admits that it would

be difficult to categorize debris solely based on captured light intensity because a

large debris far away would reflect the same amount of energy as a small debris

closer to the observer. All of these EO sensors have inherent blind spots when the

target spacecraft is in between the observer and the Sun, often referred to as the

Sun vector. These EO sensors also require some form of cooperation in order to

acquire accurate range data from the target.

The most common technique for collecting accurate range measurements

without the need for cooperation is a light detection and ranging (LIDAR) sensor

[47, 48]. A LIDAR sensor deploys a laser beam aimed at the target, and the sensor

30

measures the time it takes the reflected beam to return to the observer. Knowing

that the beam travels at the speed of light, the distance traveled can be calculated

by measuring the time traveled round trip between the target and the observer. The

laser range finder is highly accurate because it does not experience atmospheric

scattering in space, and its sensor bandpass is extremely narrow centered on the

laser’s nearly monochromatic wavelength. The effectiveness of a LIDAR device on a

non-cooperative spacecraft lies in the reflectivity of the target’s material at the

laser’s wavelength. However, with a narrow spectral bandwidth and a high powered

laser, the sensor should still detect the laser light reflected off the target without a

large gap in data caused by the Sun vector [48]. Once a range measurement is

confirmed, the range rate measurement is collected by examining the Doppler effect

of the reflected wavelength of the laser beam compared to the wavelength of the

transmitted laser beam [49].

For the non-cooperative space-based RPO scenarios in this study, a realistic

sensor suite for an observer collecting range, range rate, and angles data effectively

would be a combination of EO and LIDAR sensors. In a realistic scenario, an EO

sensor would sweep over a large area until the reflected light from the target

generates relative angle measurements from the observer. As the angle

measurements increased in accuracy, the LIDAR sensor would be able to effectively

point and follow the target, collecting range and range rate data. This study

assumes full measurement knowledge of the target in terms of range, range rate, and

angles data, which implies the initial orbit determination (IOD) on the target is

complete. Although this assumption is highly sensitive to the ability of the observer

to collect accurate and abundant measurements, an in-depth observability analysis

is not performed in this study and is left as future work.

31

2.4.3 Measurement Noise

One of the defining characteristics in assessing algorithm performance is

measurement noise. Applying the current study to the USAF SSA mission requires

accurate estimates in space-based sensor performance. Modern laser range finder

technology developed by Hablani for applications in spacecraft relative navigation

and rendezvous is used as a baseline for realistic assumptions regarding

measurement accuracy and noise [47]. These measurement accuracy values are

compared to previous literature on relative spacecraft estimation [13]. The

measurement noise will be assumed constant throughout each individual scenario,

but the process noise of the algorithm will be estimated by each filter continuously.

Error and noise estimation is outlined in greater detail in Chapter 3.

2.5 Summary

This chapter conducted a review of past and current research efforts in the

areas of orbit determination, estimation theory, adaptive estimation, and relative

satellite motion. Background for the theory necessary to set up the maneuver

detection problem was outlined. Basic derivations of the general equations used in

this study were investigated and presented. Conclusions from the research

completed on the current applications of adaptive estimation algorithms show that

the scenarios developed in this study are unique and will provide a positive

contribution to the areas of study reviewed in this chapter. Further development of

the equations and algorithms used in this study, including specific applications to

maneuver detection scenarios, are discussed in Chapter 3.

32

III. Methodology

This study focuses on investigating adaptive estimation techniques to detect

and characterize spacecraft maneuvers given a set of space-based measurements.

Section 1.2 lists the relevant research questions that will be addressed during this

research. The following chapter outlines the specific procedures and algorithms to

be used, how data will be collected through simulated scenarios, and what analysis

criteria will imply success regarding the research questions defined in this study.

3.1 Research Questions Reviewed

The research questions answered by this study transform the complex problem

of detecting non-cooperative maneuvers using adaptive estimation into a scoped and

logical path forward. The research questions encompass all aspects of assessing a

newly implemented algorithm, including efficacy, performance, and limitations. The

first step before research can begin is to analyze the methodology behind how each

research question can and will be answered to the fullest ability of this study.

The first research question addresses the first and foremost problem when

assessing a new algorithm: efficacy. Does each IMM estimation algorithm work as

anticipated, and what are specific requirements placed on each algorithm in order to

guarantee success? For this first problem, realistic parameters in each maneuver

detection scenario, such as availability of measurement data, will be adjusted for the

sake of efficacy. Scenario parameters will continue to have fewer simplifying

assumptions as the research progresses to assess performance and failure modes.

The MMAE algorithm that detects a spacecraft maneuver for the simplest case is

sufficient for completing the first research question, but success in the simplest case

may not provide adequate data with regards to performance.

33

The second research question lists specific variables that will be used to assess

the performance of each adaptive estimation algorithm. Sensor source relates to

different types of measurement data available in an RPO scenario as well as realistic

estimates for measurement noise to be implemented based on the current

space-based sensor technology available. Hablani’s patent for a space-based laser

range finder has a convenient table for the expected level of noise in the output of

the sensor [47], which will produce range, azimuth, and range rate data on the

target satellite.

Maneuver magnitudes refers to the different types of space-based propulsion

devices that produce many different levels of thrust. While a large solid rocket has

the capability of inserting a satellite into a completely new orbit regime, other

satellite missions utilize electric, cold gas, or liquid propellants for simple orbit

maintenance or attitude determination. Table 2 lists a set of common propulsion

systems used in orbit along with typical ranges for specific impulse and thrust [50].

Table 2. Common space-based propulsion systems and their applications

Propulsion System Typical Isp Range Nominal ThrustCommonApplications

Cold Gas 45-73 s 0.05-3.5 Norbital maintenanceand maneuvering;attitude control

Solid 290-304 s 25-80 kN orbit insertion

Liquid Monopropellant 200-235 s 1.5-445 Norbital maintenanceand maneuvering;attitude control

Liquid Bipropellant 274-467 s 0.1-100 kN

orbit insertion;orbital maintenanceand maneuvering;attitude control

Electric 500-3,000 s 20-2,000 mNorbit maintenanceand maneuvering;attitude control

34

The data presented in Table 2 will provide a realistic set of maneuver

magnitudes to be utilized in developing scenarios to test each adaptive estimation

algorithm. Nominal thrust calculations for space-based propulsion systems are

measured in Newtons, which allows the spacecraft user to calculate an applied force

on the spacecraft. However, the dynamical equations used to propagate the

estimation algorithms, as seen in Equation (2.15), are the second time derivative of

the state vector, which conceptually is the acceleration of the target with respect to

the observer. When estimating the thrust vector throughout each maneuver

detection scenario, it is not in fact the applied force on the target spacecraft but the

perceived acceleration of the target in the LVLH coordinate frame centered on the

observer. Given mass and propulsion information about the target, an applied

thrust could be derived from the maneuver magnitudes characterized in each

scenario.

The third parameter that will be used for assessing algorithm performance is

the relative trajectory of the target satellite. As stated in Section 2.3.2, the HCW

equations of motion assume that the deputy satellite is relatively close to the chief

satellite [37]. Although there is no explicit distance for divergence of the dynamics,

the further away the deputy gets from the target the less accurate the dynamical

model becomes. A large relative trajectory becomes a problem for the estimator

because the process noise continues to grow as the confidence in the dynamics fades,

until the point where the covariance of the process noise no longer accurately

describes the standard deviation of the estimate from the dynamics [9].

Vallado notes that significant errors are presented for maneuver detection when

using traditional least squares or filter techniques on orbital data because of a lack

of dynamical knowledge during and post-maneuver [15]. Goff concludes that the

MMAE algorithm outperforms all other traditional estimation routines by a factor

35

of 700 in maneuver detection scenarios over the course of just one orbit [7]. This

study will assess these claims for the application of the adaptive algorithm using

space-based maneuver detection scenarios.

Much of the data that assess the performance of the adaptive estimation

algorithm also answers the third research question: at what point does the

algorithm fail to converge on a solution? Many of these limitations are answered

previously, such as the minimum thrust that is able to be detected accurately or the

maximum distance away from the target where space-based measurements become

unrealistic. Another area of performance considered for the third question is the

availability of data for the observer. Through answering the first research question,

the adaptive estimation algorithm will be proven valid with unlimited measurements

on the target at all times throughout each scenario. This simplifying assumption is

not only unchallenging but also unrealistic. Through answering the third research

question, assessments will be made on the sparsity of the measurement data while

still proving algorithm success.

3.2 Overview of the Approach

In order to sufficiently answer all parts of the research questions detailed in

Section 3.1, an in-depth analysis on how the MMAE algorithm is used in this study

must be conducted. Chapter 2 reviewed the background behind MMAE and the

foundations of the VSD filter, but the specific design of the algorithm used in this

study is presented here.

3.2.1 Kalman Filter Algorithms

As discussed in Sections 2.1.3 and 2.1.4, tracking a target in real-time with

nonlinear measurements shown in Equations (2.20) and (2.21) requires a nonlinear

36

recursive estimation algorithm. The two estimation filters explored in this study are

the EKF and the UKF. The systems are set up the same for both filters using

Equations (2.1) and (2.3) to set up the state and output vector, Equations (2.15)

and (2.17) to set up the relative satellite motion dynamics, and Equations (2.20)

and (2.21) to set up the measurement basis function.

These equations discussed in depth in Chapter 2 are sufficient to initialize and

propagate the EKF for the scenarios in this study, but some predefined weights are

required for initializing the UKF algorithm. The UKF uses (2n + 1) critical points

for n number of states in the filter to transform the mean and covariance without

the need to linearize the system. These critical points, often referred to as sigma

points for their natural distribution about the mean state estimate, have certain

weighting methods to ensure the preservation of the first two moments of the

normal distribution [21]. If weighted properly, the sigma points will affect higher

moments of the distribution but keep the mean and covariance of the estimate

intact. For this study, a symmetric weighting method is used, and the sigma points

are weighted using Equation (3.1):

w0m = w0c =

κ

n+ κ(3.1a)

wjm = wjc =

1

2 (n+ κ)for j = 1, ..., 2n (3.1b)

where κ is a scalar weighting variable and n is the number of states in the system.

As discussed in [7] and [21], the UKF can match up to fourth order terms in the

distribution if the equation κ + n = 3 is satisfied, and the UT will not work for

complex sigma points, so κ > -n is the lower bound for setting the weighting

37

variable. Therefore, for all of the orbit estimation scenarios run in this study, we

will set κ = 3 - n for n states.

An extensive step by step algorithm for both the EKF and UKF used in this

study can be found in Appendix A, and many other research in this topic derive

similar algorithms: the EKF in [7] and [11], and the UKF in [7] and [51].

3.2.2 VSD Algorithm

The adaptability of the VSD filter is pivotal in the maneuver detection process.

As stated in Section 2.2.2, the VSD filter accounts for major changes in the

dynamical system by adding or removing states from the equations of motion.

Within the scope of this study, the only states being added in the VSD filter are the

two components of the thrust vector during the maneuver of the target, assuming

in-plane motion. The algorithm also makes the assumption that the thrust vector is

constant throughout each maneuver detection scenario. This causes some inherent

error in the thrust estimate because the acceleration of the target due to a constant

thrust changes slightly over time based on the mass lost by burning propellant. For

the purposes of this study, the thrust states are assumed constant, and for most

scenarios the target is assumed to produce a constant acceleration due to thrust.

Although the fact that the target is maneuvering is tracked by the VSD filter, more

specific parameters such as an estimate for the thrust magnitude and the process

noise of the new dynamics is more accurately estimated through a multiple model

framework.

3.2.3 IMM Algorithm

The IMM algorithm is used in this study to effectively estimate the process

noise of the new dynamics applied during a detected maneuver. A more accurate

38

assessment of this process noise allows for a more accurate representation of the

thrust magnitude and the amount of time spent maneuvering. As discussed in

Section 2.2.1, the IMM framework allows a band of VSD filters with different noise

assumptions to estimate the future state of the system simultaneously. The

interacting component of the algorithm compares the estimates from every VSD in

a PDF, finding a combination of the most likely estimate based on the minimization

of the MD shown in Equation (2.11). A multivariate normal PDF is used in this

study to calculate the model likelihood, and the probability update function is

shown in Equation (3.2) [30]:

Λki = N(νki ; 0,S

ki

)=

1√|2πSki |

e−12 (νki )

T(Ski )

−1νki (3.2a)

µki =µki−1Λ

ki

N∑j=1

µji−1Λji

. (3.2b)

Here Λki is the model likelihood and µki is the model weight for each model k at time

ti. After each model weight is calculated, the weighted state, covariance, and MD at

time ti are calculated using Equation (3.3) [30].

xi =N∑k=1

µki x̂ki Ψi =

N∑k=1

µki Ψki (3.3a)

Pi =N∑k=1

µki

[P̂ki +

(x̂ki − xi

) (x̂ki − xi

)T](3.3b)

where x̂ki and P̂ki are the updated state and covariance for each model k at time ti.

This study will look at two different methods for updating the model weights for the

IMM algorithm. For the full interacting method using the VSD in real-time, the

39

mixing probabilities will be calculated and a single weighted state and covariance

will be fed into each model at every time step, as seen in Equations (3.2) and (3.3)

and in Figure 4.

Figure 4. Visual diagram of the VSD IMM algorithm

As discussed in Section 2.2.1, covariance inflation is a key requirement for

algorithm convergence whenever the VSD filter is activated pre-maneuver or

deactivated post-maneuver [11]. The VSD IMM algorithm is shown to be more

accurate than a single filter [7], but is highly sensitive to small changes in the

residual vector and the measurement covariance, which can be problematic for large

maneuvers.

Another classical MMAE technique that will be explored in this study involved

post-processing the model weights after each model filters through the scenario

individually. The model weights and update equations do not change, but the

models run separate from each other, only calculating the weighted state and

covariance a posteriori. This can prevent certain model sensitivities compared to a

full interactive method, but cannot be paired with a VSD and applied to real-time

maneuver detection [30]. Instead, this classical IMM algorithm will estimate the

thrust vector throughout the entire scenario. Although there will be some inherent

error as the models are estimating a thrust even when the target is not thrusting, if

the thrust magnitude is large enough we will be able to delineate between the thrust

40

vector outputted by the target and the ambient noise of the estimator. The full step

by step IMM algorithm used in this study can be found in Appendix A.

For the purposes of this research and relating to the IMM algorithm analysis in

[30], the classical post-processing IMM algorithm is referred to as the first

generation (1G) IMM algorithm, while the full VSD IMM algorithm is referred to as

the second generation second generation (2G) IMM algorithm. A visual

representation of the 1G IMM algorithm can be seen in Figure 5.

Figure 5. Visual diagram of the 1G IMM algorithm

As shown in both Figure 4 and Figure 5, the IMM framework utilizes five

models with process noise covariance hypotheses varying at orders of magnitude

from the adjacent models in the framework. The number of models and the

distribution of hypotheses is taken from previous research [7], and for each specific

scenario the

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AIR FORCE INSTITUTE OF TECHNOLOGY · 2018. 8. 22. · Current statistical orbit determination (SOD) techniques are able to esti-mate and track trajectories for cooperative spacecraft